Some Combinatorial and Analytical Identities
Abbreviated Journal Title
partitions; identities of Chen and Liu; Dilcher; Fu and Lascoux; Prodinger and Uchimura; Summation theorems; polynomial expansions; bibasic sums; Watson transformation; the Gasper identity; Lagrange type; interpolation; Q-TAYLOR THEOREMS; DIVISOR FUNCTIONS; INTERPOLATION; FOUNDATIONS; Mathematics, Applied
We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced (4)phi(3) to a very-well-poised (8)phi(7) is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the (8)phi(7) summation theorem.
Annals of Combinatorics
"Some Combinatorial and Analytical Identities" (2012). Faculty Bibliography 2010s. 2791.